Math 131C :
Basic exam "bootcamp"
- Course description: Review of undergraduate topics in linear
algebra, analysis, and applied mathematics for the UCLA
Basic Examination.
- ANNOUNCEMENTS: None yet.
- Instructor: Zubin Gautam, MS 6603
- Lecture: T 11:00 - 11:50, R 11:00 - 12:50 MS 6118
- Office Hours: M 12:00 - 12:50, R 5:15 - 6:05
- Grading scheme: Some combination of homework, presented
problems, and exam(s) to be determined.
- Recommended texts (optional):
Linear algebra:
- Peter Petersen's lecture notes (http://www.math.ucla.edu/~petersen)
- L. Smith, Linear Algebra
- S. Lang, Linear Algebra
- S. Friedberg, A. Insel, and L. Spence, Linear Algebra
Analysis and advanced calculus:
- T. Gamelin and R. Greene, Introduction to Topology
- W. Rudin, Principles of Mathematical Analysis
- T. Tao, Analysis I and Analysis II
- C.H. Edwards, Advanced Calculus of Several Variables
Homework assignments
Topics covered
- Week 1: Basic definitions of linear algebra (vector spaces
over fields, subspaces, spanning, linear independence, bases), linear
transformations, matrix representations of linear transformations, vector
space/algebra isomorphism of spaces of linear transformations with
appropriate spaces of matrices, the Rank-Nullity Theorem. Sample Basic
problems: F05#9, W02#8.
- Tuesday, 1/15: Definition of a metric space, open and closed
sets, sequences and limit points, Cauchy sequences and completeness,
compactness, equivalent formulations of compactness in metric spaces (open
cover definition, Bolzano-Weierstrass (sequential compactness),
Heine-Borel (complete and totally bounded)). Sample Basic problem:
F01#1.
- Thursday, 1/17: Basic definitions of inner product space
theory, orthogonality, orthonormal bases and the Gram-Schmidt
construction, transposes/adjoints of matrices, adjoint operators,
symmetric/self-adjoint matrices and operators, orthogonal/unitary matrices
and operators, dual spaces, canonical dual basis, reflexivity, annihilator
of a subspace, transpose of a linear transformation, finite-dimensional
Fredholm Alternative (dual space and inner product space formulations).
- Tuesday, 1/22: Eigenvalues and eigenvectors, characteristic
polynomial, the Spectral Theorem. Sample Basic problems: F01#10, S05LA#3
- Thurday, 1/24: Equicontinuity, the Lebesgue covering lemma,
the Arzelà-Ascoli Theorem. Sample Basic problems: S05#6, F03#2.
- Tuesday, 1/29: The Stone-Weierstrass Theorem. Sample Basic
problem: F07#5a
- Thursday, 1/31: Riemann integration. Sample Basic problems:
F07#5b, F07#11, F07#12
- Tuesday, 2/5: Taylor's Theorem with remainder, Fundamental
Theorem of Calculus, numerical integration (midpoint rule). Sample Basic
problem: F07#4.
- Thursday, 2/7: Fundamental Theorem of Calculus (part 2),
differentiation under the integral sign, Banach Fixed Point Theorem
(contraction mapping principle), equivalence of norms on
finite-dimensional vector spaces. Sample Basic problems: S05A#1,
S05A#2, S04#6.
- Tuesday, 2/12: Vector-valued functions of several variables,
the derivative as a linear transformation, Jacobian matrix, partial
derivatives, directional derivatives, equality of mixed partials.
- Thursday, 2/14: Mean Value Theorem and Taylor's Theorem
(multivariable setting), Inverse Mapping Theorem. Sample Basic problem:
F03#6.
- Tuesday, 2/26: Implicit Mapping Theorem.
- Thursday, 2/28: Minimal polynomial of a linear
transformation/matrix, rational canonical form, Jordan canonical form.
- Tuesday, 3/4: Matrix exponential, application to
constant-coefficient systems of linear ordinary differential equations.
Files
- Notes on inner product
spaces
and spectral theory.
- Solution to Fall 2003 #7.